Olympiad problems

1972 Bulgaria National Olympiad, Problem 6

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that: (a) the four altitudes of $ABCD$ intersects at a common point $H$; (b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$. [i]H. Lesov[/i]

2013 Albania Team Selection Test, 4

Tags: circumcircle , trigonometry , geometry

It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$. If $AO=AH$ find the angle $\hat{BAC}$ of that triangle.

2011 NIMO Summer Contest, 15

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Let \[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ]. \] Find the remainder when $N$ is divided by 1000. [i]Proposed by Lewis Chen [/i]

2021 China Second Round, 4

Tags: combinatorics

Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying $$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$

2024 German National Olympiad, 2

Tags: 3d geometry , geometry

Six quadratic mirrors are put together to form a cube $ABCDEFGH$ with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex $A$ in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube. For each positive integer $n$, determine the set of vertices where the laser beam can leave the cube after exactly $n$ reflections.

2018 CCA Math Bonanza, L5.3

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Choose an integer $n$ from $1$ to $10$ inclusive as your answer to this problem. Let $m$ be the number of distinct values in $\left\{1,2,\ldots,10\right\}$ chosen by all teams at the Math Bonanza for this problem which are greater than or equal to $n$. Your score on this problem will be $\frac{mn}{15}$. For example, if $5$ teams choose $1$, $2$ teams choose $2$, and $6$ teams choose $3$ with these being the only values chosen, and you choose $2$, you will receive $\frac{4}{15}$ points. [i]2018 CCA Math Bonanza Lightning Round #5.3[/i]

2024 India IMOTC, 2

Tags: inequalities

Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that \[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\] [i]Proposed by Shantanu Nene[/i]

2017 IMAR Test, 3

Tags: combinatorics

We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.

2018 District Olympiad, 3

Tags: limit , convergence , real analysis , sequence

Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.

2018 ISI Entrance Examination, 5

Tags: calculus

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x\in\mathbb{R}$, $$0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}$$ Define a sequence of real numbers $\{a_n\}_{n\in\mathbb{N}}$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$ Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$, $$\vert a_n\vert \leqslant M$$

1959 AMC 12/AHSME, 47

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Assume that the following three statements are true: $I$. All freshmen are human. $II$. All students are human. $III$. Some students think. Given the following four statements: $ \textbf{(1)}\ \text{All freshmen are students.}\qquad$ $\textbf{(2)}\ \text{Some humans think.}\qquad$ $\textbf{(3)}\ \text{No freshmen think.}\qquad$ $\textbf{(4)}\ \text{Some humans who think are not students.}$ Those which are logical consequences of I,II, and III are: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 2,3\qquad\textbf{(D)}\ 2,4\qquad\textbf{(E)}\ 1,2 $

2013 Estonia Team Selection Test, 1

Tags: number theory , system of equations , diophantine , diophantine equation

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

1975 AMC 12/AHSME, 12

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If $ a \neq b$, $ a^3 \minus{} b^3 \equal{} 19x^3$, and $ a\minus{}b \equal{} x$, which of the following conclusions is correct? $ \textbf{(A)}\ a\equal{}3x \qquad \textbf{(B)}\ a\equal{}3x \text{ or } a \equal{} \minus{}2x \qquad$ $ \textbf{(C)}\ a\equal{}\minus{}3x \text{ or } a \equal{} 2x \qquad \textbf{(D)}\ a\equal{}3x \text{ or } a\equal{}2x \qquad \textbf{(E)}\ a\equal{}2x$

1993 All-Russian Olympiad, 4

Tags: function , ceiling function , combinatorics

Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?

2004 Purple Comet Problems, 18

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Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}42\\23\\\hline 65\end{tabular}\,.\]

2002 AMC 12/AHSME, 23

Tags: geometry , trigonometry , perpendicular bisector , trig identities , law of sines , angle bisector , area of a triangle

In triangle $ ABC$, side $ AC$ and the perpendicular bisector of $ BC$ meet in point $ D$, and $ BD$ bisects $ \angle ABC$. If $ AD \equal{} 9$ and $ DC \equal{} 7$, what is the area of triangle $ ABD$? $ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 14\sqrt5 \qquad \textbf{(E)}\ 28\sqrt5$

2008 Thailand Mathematical Olympiad, 7

Tags: diophantine , diophantine equation , lcm , gcd

Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

1936 Moscow Mathematical Olympiad, 027

Tags: algebra , system of equations , parameter

Solve the system $\begin{cases} x+y=a \\ x^5 +y^5 = b^5 \end{cases}$

2023 Belarus Team Selection Test, 4.3

Tags: inequalities , algebra

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

2009 Belarus Team Selection Test, 2

Tags: pentagon , equal segments , geometry

Does there exist a convex pentagon $A_1A_2A_3A_4A_5$ and a point $X$ inside it such that $XA_i=A_{i+2}A_{i+3}$ for all $i=1,...,5$ (all indices are considered modulo $5$) ? I. Voronovich

2022 Thailand Online MO, 9

Tags: number theory

The number $1$ is written on the blackboard. At any point, Kornny may pick two (not necessary distinct) of the numbers $a$ and $b$ written on the board and write either $ab$ or $\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}$ on the board as well. Determine all possible numbers that Kornny can write on the board in finitely many steps.

1973 Czech and Slovak Olympiad III A, 4

Tags: algebra , floor function , sum , sum of roots , function

For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]

2013 National Chemistry Olympiad, 56

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All of the following are condensation polymers except: $ \textbf{(A) }\text{Nylon} \qquad\textbf{(B) }\text{Polyethylene}\qquad\textbf{(C) }\text{Protein} \qquad\textbf{(D) }\text{Starch}\qquad $

2014 CHMMC (Fall), 10

Tags: combinatorics , analytic geometry

Consider a grid of all lattice points $(m, n)$ with $m, n$ between $1$ and $125$. There exists a “path” between two lattice points $(m_1, n_1)$ and $(m_2, n_2)$ on the grid if $m_1n_1 = m_2n_2$ or if $m_1/n_1 = m_2/n_2$. For how many lattice points $(m, n)$ on the grid is there a sequence of paths that goes from $(1, 1)$ to $(m, n$)?

CIME II 2018, 14

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Positive rational numbers $x<y<z$ sum to $1$ and satisfy the equation $$(x^2+y^2+z^2-1)^3+8xyz=0.$$ Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m+n < 1000$, find the maximum value of $m+n$. [I]Proposed by [b] Th3Numb3rThr33 [/b][/I]